{"id":10306,"date":"2024-11-21T15:40:16","date_gmt":"2024-11-21T14:40:16","guid":{"rendered":"https:\/\/www.mathweb.fr\/euclide\/?p=10306"},"modified":"2024-11-21T15:40:23","modified_gmt":"2024-11-21T14:40:23","slug":"approximation-dune-aire-methode-des-trapezes","status":"publish","type":"post","link":"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/","title":{"rendered":"Approximation d&#8217;une aire: m\u00e9thode des trap\u00e8zes"},"content":{"rendered":"\n<p>Pour obtenir une approximation d&#8217;une aire sous une courbe, on peut utiliser la m\u00e9thode des trap\u00e8zes.<\/p>\n\n\n\n<p>En France, la m\u00e9thode des rectangles est vaguement abord\u00e9e au programme de math\u00e9matiques en classe de terminale. Mais ce n&#8217;est pas la m\u00e9thode la plus int\u00e9ressante, loin de l\u00e0!<\/p>\n\n\n\n<!--more-->\n\n\n\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 counter-hierarchy ez-toc-counter ez-toc-white ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Au menu sur cette page...<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Approximation_dune_aire_sous_une_courbe_par_la_methode_des_trapezes_introduction\" >Approximation d&#8217;une aire sous une courbe par la m\u00e9thode des trap\u00e8zes: introduction<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Un_exemple_avec_n_6\" >Un exemple avec n = 6<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Approximation_de_laire_sous_la_courbe\" >Approximation de l&#8217;aire sous la courbe<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Calcul_a_laide_de_Python\" >Calcul \u00e0 l&#8217;aide de Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Calcul_de_laire_avec_integrale\" >Calcul de l&#8217;aire avec int\u00e9grale<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Generalisation_dune_formule_de_la_methode_des_trapezes\" >G\u00e9n\u00e9ralisation d&#8217;une formule de la m\u00e9thode des trap\u00e8zes<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Premier_code_en_Python\" >Premier code en Python<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/#Second_code_en_Python\" >Second code en Python<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Approximation_dune_aire_sous_une_courbe_par_la_methode_des_trapezes_introduction\"><\/span>Approximation d&#8217;une aire sous une courbe par la m\u00e9thode des trap\u00e8zes: introduction<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Nous allons voir comment approximer l&#8217;aire du domaine d\u00e9limit\u00e9 par la courbe repr\u00e9sentative d&#8217;une fonction <em>f<\/em> continue sur un intervalle [<em>a<\/em> ; <em>b<\/em>] et l&#8217;axe des abscisses.<\/p>\n\n\n\n<p>Pour cela, \u00e0 l&#8217;instar de la m\u00e9thode des rectangles, on commence par subdiviser l&#8217;intervalle [<em>a<\/em> ; <em>b<\/em>] en <em>n<\/em> subdivisions de m\u00eame amplitude. Ensuite, on note:$$\\forall~ 0\\leq  k \\leq n,\\quad x_k=a+k\\frac{b-a}{n}.$$<\/p>\n\n\n\n<p>On consid\u00e8re alors les trap\u00e8zes \\(T_k\\) de bases \\(f(x_k)\\) et \\(f(x_{k+1})\\). <\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Un_exemple_avec_n_6\"><\/span>Un exemple avec n = 6<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>C&#8217;est un peu indigeste \u00e9crit comme cela, donc on va regarder ce que cela donne:<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium\"><a href=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-1.png\" data-fancybox=\"gallery\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"239\" src=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-1-300x239.png\" alt=\"approximation d'une aire par la m\u00e9thode des trap\u00e8zes\" class=\"wp-image-10307\" srcset=\"https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-1-300x239.png 300w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-1-768x611.png 768w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-1-600x477.png 600w, https:\/\/www.mathweb.fr\/euclide\/wp-content\/uploads\/2024\/11\/image-1.png 810w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/a><\/figure>\n<\/div>\n\n\n<p>J&#8217;ai ici pris l&#8217;exemple de la fonction$$f(x)=\\frac{1}{4}(x-4)^3-2(x-4)+3$$ et j&#8217;ai consid\u00e9r\u00e9 l&#8217;intervalle [1 ; 7] que j&#8217;ai subdivis\u00e9 en 6 intervalles d&#8217;amplitude 1.<\/p>\n\n\n\n<p>Rappelons que l&#8217;aire d&#8217;un trap\u00e8ze est donn\u00e9e par la formule:$$\\frac{(B+b)\\times h}{2}$$o\u00f9:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><em>B<\/em> est la longueur de la grande base,<\/li>\n\n\n\n<li><em>b<\/em> est celle de la petite base,<\/li>\n\n\n\n<li><em>h<\/em> est la hauteur du trap\u00e8ze.<\/li>\n<\/ul>\n\n\n\n<p>Le trap\u00e8ze \\(T_k\\) a donc pour aire:$$\\mathcal{A}_k=\\frac{1}{2}\\Big[ \\big( f(x_k)+f(x_{k+1}) \\big) \\times 1 \\Big].$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Approximation_de_laire_sous_la_courbe\"><\/span>Approximation de l&#8217;aire sous la courbe<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Il suffit maintenant d&#8217;ajouter l&#8217;aire de tous les trap\u00e8zes pour avoir une approximation (tr\u00e8s grossi\u00e8re) de l&#8217;aire qui nous int\u00e9resse: $$\\sum_{k=0}^6 \\mathcal{A}_k = \\frac{1}{2}\\Big[ f(x_0) + 2f(x_1) + 2f(x_2) + \\cdots + 2f(x_5) + f(x_6) \\Big].$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Calcul_a_laide_de_Python\"><\/span>Calcul \u00e0 l&#8217;aide de Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Ici, trois lignes suffisent \u00e0 calculer cette derni\u00e8re somme:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">f = lambda x: 0.25*(x-4)**3 - 2*(x-4) + 3\naire = 0.5 * sum( [ f(1) , f(7) ] + [ 2*f(x) for x in range(2,7) ] )\nprint( aire )<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>>>> 18.0<\/code><\/pre>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Calcul_de_laire_avec_integrale\"><\/span>Calcul de l&#8217;aire avec int\u00e9grale<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Notre aire se calcule \u00e0 l&#8217;aide de l&#8217;int\u00e9grale:$$\\int_1^7 f(x)\\text{d}x.$$<\/p>\n\n\n\n<p>Pour calculer cette int\u00e9grale on cherche une primitive de:$$f(x)=\\frac{1}{4}x^3-3x^2+10x-5.$$ On trouve alors: $$F(x)=\\frac{1}{16}x^4-x^3+5x^2-5x.$$<\/p>\n\n\n\n<p>Maintenant, on a:$$\\int_1^7 f(x)\\text{d}x = F(7)-F(1)=17,0625 &#8211; (-0,9375) = 18.$$<\/p>\n\n\n\n<p>On obtient exactement la m\u00eame valeur que celle obtenue \u00e0 l&#8217;aide de l&#8217;approximation pr\u00e9c\u00e9dente&#8230; Mais attention! Ceci est exceptionnel&#8230; En g\u00e9n\u00e9ral, notre m\u00e9thode des trap\u00e8zes ne donne r\u00e9ellement qu&#8217;une <em>approximation<\/em>.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Generalisation_dune_formule_de_la_methode_des_trapezes\"><\/span>G\u00e9n\u00e9ralisation d&#8217;une formule de la m\u00e9thode des trap\u00e8zes<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n\n\n<p>Nous allons reprendre le raisonnement pr\u00e9c\u00e9dent en g\u00e9n\u00e9ralisant.<\/p>\n\n\n\n<p><strong>Dans un premier temps,<\/strong> nous subdivisons l&#8217;intervalle [<em>a<\/em> ; <em>b<\/em>] en <em>n<\/em>: la hauteur des trap\u00e8zes sera alors \u00e9gale \u00e0 \\(\\frac{b-a}{n}\\).<\/p>\n\n\n\n<p><strong>Ensuite,<\/strong> l&#8217;aire du trap\u00e8ze \\(T_k\\) sera \u00e9gale \u00e0 : \\( \\mathcal{A}_k = \\frac{1}{2} \\big(f(x_k)+f(x_{k+1})\\big)\\times\\frac{b-a}{n}\\).<\/p>\n\n\n\n<p><strong>On en conclut alors<\/strong> que la somme des aires des trap\u00e8zes est:$$\\begin{array}{ll}\\displaystyle\\sum_{k=0}^n \\mathcal{A}_k &amp; = \\displaystyle\\frac{1}{2}\\times\\frac{b-a}{n}\\sum_{k=0}^n \\big(f(x_k)+f(x_{k+1})\\big) \\\\ &amp; = \\frac{b-a}{2n}\\Big[ f(x_0)+f(x_{n+1}) + 2\\big( f(x_1)+f(x_2)+\\cdots+f(x_n) \\big)\\Big]\\end{array}$$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Premier_code_en_Python\"><\/span>Premier code en Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<p>Si l&#8217;on souhaite un programme court, mais peu lisible, on peut opter pour le code suivant:<\/p>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">f = lambda x: 0.25*(x-4)**3 - 2*(x-4) + 3\nn = 500\na, b = 1, 7\naire = ( (b-a) \/ (2*n) ) * sum( [ f(a) , f(b) ] + [ 2*f(a+k*(b-a)\/n) for k in range(n) ] )\nprint( aire )<\/pre>\n\n\n\n<p>Mais je suis conscient qu&#8217;un tel programme n&#8217;est pas tr\u00e8s digeste. Je vous propose alors le code suivant.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Second_code_en_Python\"><\/span>Second code en Python<span class=\"ez-toc-section-end\"><\/span><\/h3>\n\n\n\n<pre class=\"EnlighterJSRAW\" data-enlighter-language=\"python\" data-enlighter-theme=\"dracula\" data-enlighter-highlight=\"\" data-enlighter-linenumbers=\"\" data-enlighter-lineoffset=\"\" data-enlighter-title=\"\" data-enlighter-group=\"\">def f(x):\n    return 0.25*(x-4)**3 - 2*(x-4) + 3\n\ndef trapezoid_method(a,b,n):\n    h = (b - a) \/ n # hauteur des trap\u00e8zes\n    aire = 0\n    for k in range(n):\n        x1 = a + k*h\n        x2 = a + (k+1)*h\n        aire = aire + h*(f(x1) + f(x2))\/2\n        \n    return aire<\/pre>\n\n\n\n<pre class=\"wp-block-code\"><code>>>> trapezoid_method(1,7,500)\n18.00000000000001\n\n>>> trapezoid_method(1,7,50)\n18.000000000000007\n\n>>> trapezoid_method(1,7,10)\n18.0\n\n>>> trapezoid_method(1,7,5)\n18.0<\/code><\/pre>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Pour obtenir une approximation d&#8217;une aire sous une courbe, on peut utiliser la m\u00e9thode des trap\u00e8zes. En France, la m\u00e9thode des rectangles est vaguement abord\u00e9e au programme de math\u00e9matiques en classe de terminale. Mais ce n&#8217;est pas la m\u00e9thode la plus int\u00e9ressante, loin de l\u00e0!<\/p>\n","protected":false},"author":1,"featured_media":10314,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,5],"tags":[],"class_list":["post-10306","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematiques","category-python"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.5 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Approximation d&#039;une aire: m\u00e9thode des trap\u00e8zes - Mathweb.fr<\/title>\n<meta name=\"description\" content=\"Pour avoir une approximation d&#039;une aire, on peut utiliser la m\u00e9thode des trap\u00e8zes, qui s&#039;av\u00e8re meilleure que celle des rectangles.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.mathweb.fr\/euclide\/2024\/11\/21\/approximation-dune-aire-methode-des-trapezes\/\" \/>\n<meta property=\"og:locale\" content=\"fr_FR\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Approximation d&#039;une aire: m\u00e9thode des trap\u00e8zes - 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